Problem: Rewrite as a simplified fraction. ${2.13\overline{7} = {?}}$
Solution: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 2137.7777...\\ 100x &= 213.7777...\end{align*}}$ Subtracting the two equations, we have: ${900x = 1924}$ ${x = \dfrac{1924}{900}} $ ${x = \dfrac{481}{225}} \text{or} {2 \dfrac{31}{225}}$